
theorem Th23:
  for V be non empty ModuleStr over F_Complex, f be Functional of
  V holds ker f = ker f*'
proof
  let V be non empty ModuleStr over F_Complex, f be Functional of V;
  thus ker f c= ker f*'
  proof
    let x be object;
    assume x in ker f;
    then consider v be Vector of V such that
A1: x=v and
A2: f.v = 0.F_Complex;
    (f*').v = 0.F_Complex by A2,Th22;
    hence thesis by A1;
  end;
  let x be object;
  assume x in ker f*';
  then consider v be Vector of V such that
A3: x=v and
A4: (f*').v = 0.F_Complex;
  f.v = 0.F_Complex by A4,Th22;
  hence thesis by A3;
end;
